Function of A Complex Variable

[darkchild]

Homework Statement

Ifz=e^{2\pi i/5}, then 1+z+z^{2}+z^{3}+5z^{4}+4z^{5}+4z^{6}+4z^{7}+4z^{8}+5z^{9}=

(A) 0

(B) 4e^{3\pi i/5}

(C) 5e^{4\pi i/5}

(D) -4e^{2\pi i/5}

(E)-5e^{3\pi i/5}

Homework Equations

e^{2\pi i}=\cos(2\pi)+isin(2\pi)=1

The Attempt at a Solution

I plugged z=1 into the equation and calculated 30. None of the answer choices is equal to 30. I'm thinking that maybe I have to do something with the fifth roots of unity, but I'm not sure what.

 

[Dick]

You might want to use that the sum of the five fifth roots of unity is zero.

 

[spamiam]

Studying for the Math GRE, eh?

Anyway, I was just going to add that the fact that Dick mentioned comes from the factorization of z^5 - 1. Letting \zeta = e^{{2 \pi i}/{5}} (I want z for my variable), which is a primitive 5th root of unity, we have

<br /> z^5 - 1 = (z - 1)(z- \zeta)(z - \zeta^2)(z - \zeta^3)(z - \zeta^4)<br />

Calculating the coefficient for z⁴ yields the identity.